I had the same problem until I read "The Lost Art of House Cleaning" by Jan M. Dougherty. She is a professional house cleaner and this is one of my favorite all-time books. This is her method which I use after having tried various expensive machines.
1. Vacuum the edges where the floor meets the baseboard, then vacuum the rest of the floor.
2. Spray Krud Kutter on the edge and corners. (Gallon size bottles sold at paint dept. in Home Depot, etc.) Wipe with a white terry cloth rag. Use a detail brush if wiping with the rag doesn't get it clean.
3. Mop the rest of the floor by spraying with Krud Kutter and mopping with a white terry cloth rag on a Swiffer mop. You can buy white terry cloth rags in large packages at Sams, Target, etc. usually found in the auto dept. Use the rags with abandon and change them whenever they get dirty. Keep spraying and mopping until the rag comes up clean. You will need to use Krud Kutter full strength the first time you clean the floor, then you can use it diluted at a 5:1 ratio.
4. After the floor is clean, finish by spraying the tile with vinegar, and mopping with a white terry cloth rag on a Swiffer.